Complex-number analytic functions {elliptic function} can be doubly periodic, be single-valued, have poles, and have singularity at infinity. Elliptic functions are power-series-equation solutions. They are elliptic-integral inverses. They are complex functions, because period ratios cannot be real numbers.
Weierstrass
Differential equations can have form dx^2 / dt^2 = A + B * x + C * x^2, where x is complex number. It has integral solutions {Weierstrass elliptic function}.
Jacobi
Differential equations can have form dx^2 / dt^2 = A + B * x + C * x^2 + D * x^3, where x is complex number. It has integral solutions {Jacobi elliptic function}.
Elliptic functions {Jacobian functions} can have higher powers. Determinants give solutions.
Darboux
Integrals {elliptic integral} {Darboux integral} can be (integral of P(x)) / (R(x))^0.5, where P is rational function, R is fourth-degree polynomial, and x is complex variable.
Abelian
Elliptic functions {Abelian integral} can have integral additions. They can also have algebraic and logarithmic terms. Integral of (R(u, z)) * dz, where f(u, z) = 0, requires more than one integral to describe domain. For Abelian integrals, equation genus is number of integrals needed to express solution {Abel's theorem}.
Mathematical Sciences>Algebra>Function>Kinds>Complex
3-Algebra-Function-Kinds-Complex
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Date Modified: 2022.0224